The problem is not new. Several disputes on the subject of rebirth are recorded in traditional texts. In Pāḷi we have the Katthāvatthu - a record of various intellectual disputes with other Buddhists by the Theravādins. In Sanskrit there is Nāgārjuna's Mūlamadhyamakārikā (MMK) which disputes the Mainstream Buddhist view of karma and rebirth. Vasubandhu's Abhidharmakośa records disputes between Sarvāstivādins, Sautrantikas, and others on this subject.
There are a number of points of dispute, but two principle and related arguments. These arguments assume that rebirth is what happens after we die, and that karma is the driving force behind rebirth - or in other words, that rebirth is the primarily effect (vipāka) of actions (karma). It's not well understood that the early Buddhists texts in particular do not teach "actions have consequences" so much as they teach "actions cause rebirth".
Firstly there is the problem of action at a temporal distance. I first described this problem in an essay called "Does Karma Break the Rules." The basic question here is, how does karma manifest an effect (vipāka) long after the cessation of the condition? This is specifically denied by the principle of dependent arising (pratītyasamutpāda), which says that when a condition ceases the effect must also cease, with the implication (in the grammar of the statement) that this happens either simultaneously or in immediate sequence. There is no question of a delayed effect from a condition in dependent arising. This argument is made by Nāgārjuna, for example, in chp 17 of MMK:
If the action endures to the time of maturation, then it would permanent
If it is destroyed, then being destroyed, what fruit will it produce?
(MMK 17.6. Siderits and Katsura, p.176)
Since the main "effect" of karma is precisely rebirth, then this is a very important question to decide. Nāgārjuna answers it by saying that:
Defilements, actions, and bodies, agents, and fruits, are similar to the city of gandharvas; they are like a mirage, a dream (MMK 17.33, p.191)
But this answer was not routinely accepted by other Buddhists, because even though it solves the problem on the surface, the implications are that nothing is real and most Buddhists found this unsatisfactory. Nāgārjuna was accused of being a nihilist. And in any case the Theravādins ignored these developments in India.
Three main theories emerged to try to show how karma might cause rebirth. The doctrine of momentariness, the doctrine of always existent (sarva-asti) dharmas, and the seed theory.
In momentariness, adopted by the Theravādins, each karma is conceived of as a citta (In AN 6.63 the Buddha says cetanāhaṃ kammaṃ vadāmi - "intention is what I call karma"). This citta arises and passes away in a very short space of time (a moment) and is the condition for the arising of a second identical citta. That citta arises and passes away and is the condition for a third citta. And so on until the final effect of the action manifests, usually combined with other cittas to produce a final citta at death (cuticitta) that conditions the first citta of a new life elsewhere (paṭisandhicitta).
The always existent theory, associated with the Sarvāstivāda, argues that if a dharma (i.e. a mental event) has an effect in the future, then it must exist in the future, else it violates pratītyasamutpāda. Similarly if a past dharma has an effect in the present, then it must still exist in the past or it violates pratītyasamutpāda. Thus dharmas must always exist, in some form, but may only be "active" in the present - which is why we only perceive them in the present. This is a much better theory than most Buddhists give them credit for. Most other Buddhists modified dependent arising to allow for karma, but the Sarvāstivādins opted to retain dependent arising unchanged and modified dharmas. Of course other schools rejected this as a form of eternalism.
The seed theory starts off as a metaphor: cittas are like seeds that are planted in the soil and grow into a plant. The plant and the seed are not the same, nor do they co-exist in time, but one develops into the other. This is an elegant metaphor, but we have no idea of the underlying mechanism. Vasubandhu takes this metaphor and formalises it - this is definitely how karma and rebirth work. He introduces the idea of the ālayavijñāna "store consciousness". But his Yogācāra followers reified the metaphor and made the ālayavijñāna a real thing, thus making the theory eternalist and a wrong view.
Of the three, only 1. and 3. are currently in use, but 2. was very prominent in North India for many centuries. Each theory introduces a number of ad hoc alterations to the received tradition - indicating that they could not see how karma and rebirth worked as things stood. Unfortunately none of the new formulas really solved the problem (nor did any of the less influential attempts not mentioned here). In several articles on my blog I show that the logic of these solutions to the problem, though in some cases admirable and elegant, actually fail to explain how karma and rebirth work. See especially The Logic of Karma. This leaves us in the present with no workable theory of karma and rebirth. And this is before we invoke science!
Secondly, this led to another big argument, the one over the interim realm (antarābhava). The doctrine of momentariness requires that the stream of cittas be unbroken. For this purpose they had to invent a kind of "neutral gear" called the bhavaṅgacitta which is what the mind is doing when it is not doing anything. It primarily provides continuity when our mind is inactive - such as when we sleep or in the nirodhasamāpatti "attainment of cessation". In this case there can be no gap between death and rebirth or the stream of cittas would be broken.
On the other hand, for other Buddhists the idea of an instantaneous transition did not make sense. Some pointed out that a cart carrying rice from one village to another took an appreciable time to get there. How could a citta fly from one person to another in no time? So they had to invent the interim realm, the antarābhava or bardo in Tibetan, to accommodate this.
At most, only one of these views can be correct. There can either be an interim or not. This dispute is never resolved in Buddhism so that today the official position of the Theravāda school is that there can be no interim realm, and Tibetan schools make the bardo between death and rebirth a particular feature of their teaching. Ironically many modern Theravādins (such as Bhikkhu Sujato) find the interim realm plausible, even though it destroys the stream of cittas that underlies their view of karma and rebirth.
I have been writing essays on this subject for some years now including critiques of the traditionalist views of some prominent Buddhists. Some of my work has also been published in academic journals. A complete list can be found under the heading Afterlife.
In addition I was interviewed by Ted Meissner of The Secular Buddhist on the subject of karma and rebirth:
Doug Smith of the Secular Buddhist Association has also critiqued the Traditionalist view of rebirth: http://secularbuddhism.org/2013/05/29/a-secular-evaluation-of-rebirth/
Someone put together this discussion between me and David Chapman on Twitter on storify.